There are many ways to represent numbers. In any case, a number is represented by a symbol or a group of symbols (a word) of some alphabet. Such symbols are called numbers.
Number systems
Non-positional and positional number systems are used to represent numbers.
Non-positional number systems
As soon as people started counting, they began to need to write down numbers. Finds by archaeologists at the sites of primitive people indicate that initially the number of objects was displayed by an equal number of some kind of icons (tags): notches, dashes, dots. Later, to make counting easier, these icons began to be grouped in groups of three or five. This system of writing numbers is called unit (unary), since any number in it is formed by repeating one sign, symbolizing one. Echoes of the unit number system are still found today. So, to find out what course a military school cadet is studying in, you need to count how many stripes are sewn on his sleeve. Without realizing it, kids use the unit number system, showing their age on their fingers, and counting sticks are used to teach 1st grade students how to count. Let's consider various systems Reckoning.
The unit system is not the most convenient way to write numbers. Write it this way large quantities It's tedious, and the notes themselves turn out to be very long. Over time, other, more convenient number systems arose.
Ancient Egyptian decimal non-positional number system. Around the third millennium BC, the ancient Egyptians came up with their own numerical system, in which the key numbers were 1, 10, 100, etc. special icons were used - hieroglyphs. All other numbers were composed from these key numbers using the operation of addition. The number system of Ancient Egypt is decimal, but non-positional. In non-positional number systems, the quantitative equivalent of each digit does not depend on its position (place, position) in the number record. For example, to depict 3252, three lotus flowers (three thousand), two rolled palm leaves (two hundreds), five arcs (five tens) and two poles (two units) were drawn. The size of the number did not depend on the order in which its constituent signs were located: they could be written from top to bottom, from right to left, or interspersed.
Roman number system. An example of a non-positional system that has survived to this day is the number system that was used more than two and a half thousand years ago in Ancient Rome. The Roman number system was based on the signs I (one finger) for the number 1, V (open palm) for the number 5, X (two folded palms) for 10, and the first letters of the corresponding Latin words began to be used to designate the numbers 100, 500 and 1000 (Centum – one hundred, Demimille – half a thousand, Mille – one thousand). To write down a number, the Romans decomposed it into the sum of thousands, half thousand, hundreds, fifty, tens, heels, units. For example, the decimal number 28 is represented as follows:
XXVIII=10+10+5+1+1+1 (two tens, heels, three ones).
To record intermediate numbers, the Romans used not only addition, but also subtraction. In this case, the following rule was applied: each smaller sign placed to the right of the larger one is added to its value, and each smaller sign placed to the left of the larger one is subtracted from it. For example, IX stands for 9, XI stands for 11.
The decimal number 99 has the following representation:
XCIХ = –10+100–1+10.
Roman numerals have been used for a very long time. Even 200 years ago, in business papers, numbers had to be denoted by Roman numerals (it was believed that ordinary Arabic numerals were easy to counterfeit). The Roman numeral system is used today mainly for naming significant dates, volumes, sections and chapters in books.
Alphabetic number systems. Alphabetic systems were more advanced non-positional number systems. Such number systems included Greek, Slavic, Phoenician and others. In them, numbers from 1 to 9, whole numbers of tens (from 10 to 90) and whole numbers of hundreds (from 100 to 900) were designated by letters of the alphabet. In the alphabetic number system Ancient Greece the numbers 1, 2, ..., 9 were designated by the first nine letters of the Greek alphabet, etc. The following 9 letters were used to denote the numbers 10, 20, ..., 90, and the last 9 letters were used to denote the numbers 100, 200, ..., 900.
Among the Slavic peoples, the numerical values of the letters were established in the order of the Slavic alphabet, which used first the Glagolitic alphabet and then the Cyrillic alphabet.
In Russia, Slavic numbering was preserved until the end of the 17th century. Under Peter I, the so-called Arabic numbering prevailed, which we still use today. Slavic numbering was preserved only in liturgical books.
Non-positional number systems have a number of significant disadvantages:
- There is a constant need to introduce new symbols for recording large numbers.
- It is impossible to represent fractional and negative numbers.
- It is difficult to perform arithmetic operations because there are no algorithms for performing them.
Positional number systems
In positional number systems, the quantitative equivalent of each digit depends on its position (position) in the code (record) of the number. Nowadays we are accustomed to using the decimal positional system - numbers are written using 10 digits. The rightmost digit denotes units, the one to the left - tens, even further to the left - hundreds, etc.
For example: 1) sexagesimal (Ancient Babylon) – the first positional number system. Until now, when measuring time, a base of 60 is used (1min = 60s, 1h = 60min); 2) duodecimal number system (the number 12—“dozen”—was widely used in the 19th century: there are two dozen hours in a day). Counting not by fingers, but by knuckles. Each finger, except the thumb, has 3 joints - 12 in total; 3) currently the most common positional number systems are decimal, binary, octal and hexadecimal (widely used in low-level programming and in computer documentation in general, since modern computers The minimum unit of memory is an 8-bit byte, the values of which are conveniently written in two hexadecimal digits).
In any positional system, a number can be represented as a polynomial.
Let's show how to represent a decimal number as a polynomial:
Types of number systems
The most important thing you need to know about the number system is its type: additive or multiplicative. In the first type, each digit has its own meaning, and to read the number you need to add up all the values of the digits used:
XXXV = 10+10+10+5 = 35; CCXIX = 100+100+10–1+10 = 219;
In the second type, each digit can have different meanings depending on its location in the number:
(hieroglyphs in order: 2, 1000, 4, 100, 2, 10, 5)
Here the hieroglyph “2” is used twice, and in each case it took on different meanings “2000” and “20”.
2´ 1000 + 4´ 100+2´ 10+5 = 2425
For an additive (“additional”) system, you need to know all the numbers and symbols with their meanings (there are up to 4-5 dozen of them), and the order of recording. For example, in Latin notation, if a smaller digit is written before a larger one, then subtraction is performed, and if after, then addition (IV = (5–1) = 4; VI = (5+1) = 6).
For a multiplicative system, you need to know the image of the numbers and their meaning, as well as the base of the number system. Determining the base is very easy; you just need to recalculate the number of significant digits in the system. To put it simply, this is the number from which the second digit of the number begins. For example, we use the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. There are exactly 10 of them, so the base of our number system is also 10, and the number system is called “decimal”. The above example uses the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (auxiliary 10, 100, 1000, 10000, etc. do not count). There are also 10 main numbers here, and the number system is decimal.
As you can guess, as many numbers as there are, there can be as many number system bases. But only the most convenient bases of number systems are used. Why do you think the base of the most commonly used human number system is 10? Yes, precisely because we have 10 fingers on our hands. “But there are only five fingers on one hand,” some will say, and they will be right. The history of mankind knows examples of five-fold number systems. “And with legs there are twenty toes,” others will say, and they will also be absolutely right. This is exactly what the Mayans believed. This can even be seen in their numbers.
The concept of “dozen” is very interesting. Everyone knows that this is 12, but few people know where this number came from. Look at your hands, or rather, one hand. How many phalanges are there on all the fingers of one hand, not counting the thumb? That's right, twelve. And the thumb is intended to mark the counted phalanges.
And if on the other hand we mark the number of full dozens with our fingers, we will get the well-known sexagesimal Babylonian system.
Different civilizations counted differently, but even now you can even find in the language, in the names and images of numbers, the remains of completely different number systems that were once used by these people.
So the French once had a base-20 number system, since 80 in French sounds like “four times twenty.”
The Romans, or their predecessors, once used the fivefold system, since V is nothing more than the image of a palm with the thumb extended, and X is two of the same hands.
Non-positional number systems
People learned to count a long time ago. Subsequently, the need arose to record numbers. The number of objects was depicted by drawing dashes or notches on some hard surface. So that two people could accurately store some numerical information, they took a wooden tag, made the required number of notches on it, and then split the tag in half. Everyone took away their other half and kept it. This technique allowed us to avoid controversial situations. Archaeologists found such records during excavations. They date back to the 10-11th millennium BC.
Scientists called this system of writing numbers unit (unary), since any number in it is formed by repeating one sign symbolizing one.
Later, these badges began to be combined into groups of 3, 5 and 10 sticks. Therefore, more convenient systems Reckoning.
Around the third millennium BC, the Egyptians came up with their own numerical system, in which special icons - hieroglyphs - were used to indicate key numbers. Each such hieroglyph could be repeated no more than 9 times. This number system is called Ancient Egyptian decimal non-positional number system
An example of a non-positional number system that has survived to this day is the number system used more than two and a half thousand years ago in Ancient Rome. It is calledroman number system.
It is based on the signs I(1), V(5), X(10), L(50), C(100), D(500), M(1000).
Roman numerals have been used for a very long time; today they are used mainly to name significant dates, volumes, sections and chapters in books.
To write down numbers, the Romans used not only addition, but also subtraction.
Rules for compiling numbers in the Roman numeral system:
- Several identical numbers in a row are added together (group of the first type).
- If there is a smaller one to the left of the larger digit, then the value of the smaller digit is subtracted from the value of the larger one (group of the second type).
- The values of groups and numbers not included in the groups of the first and second types are added together.
In ancient times, number systems reminiscent of the Roman ones were widely used in Rus'. They were called yasak. With their help, tax collectors filled out tax payment receipts (yasak) and made entries in the tax notebook.
"Russian Book of Taxes"
Non-positional number systems have a number of significant disadvantages:
- There is a constant need to introduce new symbols for recording large numbers.
- It is impossible to represent fractional and negative numbers.
- It is difficult to perform arithmetic operations because there are no algorithms for performing them. In particular, all peoples, along with number systems, had methods of finger counting, and the Greeks had an abacus counting board - something like our abacus.
But we still use elements of the non-positional number system in everyday speech, in particular, we say one hundred, not ten tens, a thousand, a million, a billion, a trillion.
Unit number system
The need to write numbers began to arise among people in ancient times after they learned to count. Evidence of this is archaeological finds in the places of camps of primitive people, which date back to the Paleolithic period ($10$-$11$ thousand years BC). Initially, the number of objects was depicted using certain signs: dashes, notches, circles marked on stones, wood or clay, as well as knots on ropes.
Picture 1.
Scientists call this system of notating numbers unit (unary), since the number in it is formed by the repetition of one sign, which symbolizes one.
Disadvantages of the system:
when writing large number nessesary to use a large number of sticks;
It can be easy to make mistakes when applying sticks.
Later, to make counting easier, people began to combine these signs.
Example 1
Examples of using the unit number system can be found in our lives. For example, small children try to depict how old they are on their fingers, or counting sticks are used to teach counting in the first grade.
Unit system not entirely convenient, since the entries look very long and their writing is quite tedious, so over time, more practical number systems began to appear.
Here are some examples.
Ancient Egyptian decimal non-positional number system
This system Numbers appeared around 3000 BC. as a result of the fact that the inhabitants of Ancient Egypt came up with their own numerical system, in which when designating key numbers $1$, $10$, $100$, etc. hieroglyphs were used, which was convenient when writing on clay tablets that replaced paper. Other numbers were made from them using addition. First, the number of the highest order was written down, and then the lower one. The Egyptians multiplied and divided, successively doubling numbers. Each digit could be repeated up to $9$ times. Examples of numbers of this system are given below.
Figure 2.
Roman number system
This system is fundamentally not much different from the previous one and has survived to this day. It is based on the following signs:
$I$ (one finger) for the number $1$;
$V$ (open palm) for the number $5$;
$X$ (two folded palms) for $10$;
to denote the numbers $100$, $500$ and $1000$, the first letters of the corresponding Latin words were used ( Сentum- one hundred, Demimille- half a thousand, Mille- thousand).
When composing numbers, the Romans used the following rules:
The number is equal to the sum of the values of several identical “digits” located in a row, forming a group of the first type.
The number is equal to the difference in the values of two “digits” if the smaller one is to the left of the larger one. In this case, the value of the smaller one is subtracted from the larger value. Together they form a group of the second type. In this case, the left “digit” can be less than the right one by a maximum of $1$ order: only $X(10$) can be in front of $L(50)$ and $C(100$), among the “lowest” ones, only $X(10$) can be in front of $D(500$ ) and $M(1000$) – only $C(100$), before $V(5) – I(1)$.
The number is equal to the sum of the group values and “digits” not included in the $1$ or $2$ groups.
Figure 3.
Roman numerals have been used since ancient times: they indicate dates, numbers of volumes, sections, and chapters. I used to think that ordinary Arabic numerals could be easily faked.
Alphabetic number systems
These number systems are more advanced. These include Greek, Slavic, Phoenician, Jewish and others. In these systems, numbers from $1$ to $9$, as well as the number of tens (from $10$ to $90$), hundreds (from $100$ to $900$) were designated by letters of the alphabet.
In the ancient Greek alphabetic number system, the numbers $1, 2, ..., 9$ were represented by the first nine letters of the Greek alphabet, etc. The following $9$ letters were used to denote the numbers $10, 20, ..., 90$, and the last $9$ letters were used to denote the numbers $100, 200, ..., 900$.
Among the Slavic peoples, the numerical values of letters were established in accordance with the order of the Slavic alphabet, which initially used the Glagolitic alphabet and then the Cyrillic alphabet.
Figure 4.
Note 1
The alphabetic system was also used in ancient Rus'. Until the end of the $17th century, $27$ Cyrillic letters were used as numbers.
Non-positional number systems have a number of significant disadvantages:
There is a constant need to introduce new symbols for recording large numbers.
It is impossible to represent fractional and negative numbers.
It is difficult to perform arithmetic operations because there are no algorithms for performing them.
test
Positional and non-positional number systems
The various number systems that existed in the past and that are used today can be divided into non-positional and positional. The signs used to write numbers are called digits.
In non-positional number systems, the position of the digit in the number notation does not depend on the value it represents. An example of a non-positional number system is the Roman system, which uses Latin letters as numbers.
In positional number systems, the value denoted by a digit in a number depends on its position. The number of digits used is called the base of the number system. The place of each digit in a number is called position. The first system known to us based on the positional principle is Babylonian sexagesimal. The numbers in it were of two types, one of which denoted units, the other - tens.
Currently, positional number systems are more widespread than non-positional number systems. This is because they allow large numbers to be written using a relatively small number of characters. An even more important advantage of positional systems is their simplicity and ease of implementation. arithmetic operations over the numbers written in these systems.
The most commonly used is the Indo-Arabic decimal system. The Indians were the first to use zero to indicate the positional significance of a quantity in a string of numbers. This system is called decimal because it has ten digits.
The difference between positional and non-positional number systems is most easily understood by comparing two numbers. In the positional number system, comparison of two numbers occurs as follows: in the numbers under consideration, from left to right, digits in the same positions are compared. A larger number corresponds to a larger number value. For example, for the numbers 123 and 234, 1 is less than 2, so 234 is greater than 123. In a non-positional number system, this rule does not apply. An example of this would be the comparison of two numbers IX and VI. Even though I is smaller than V, IX is larger than VI.
The base of the number system in which a number is written is usually indicated by a subscript. For example, 555 7 is a number written in the decimal number system. If a number is written in the decimal system, then the base is usually not indicated. The base of the system is also a number, and it is indicated in the usual decimal system. Any integer in the positional system can be written in polynomial form:
Х s =(A n A n-1 A n-2 ...A 2 A 1 ) s =A n ·S n-1 +A n-1 ·S n-2 +A n-2 ·S n- 3 +...+A 2 ·S 1 +A 1 ·S 0
where S is the base of the number system, And n is the digits of the number written in this number system, n is the number of digits of the number.
So, for example, the number 6293 10 will be written in polynomial form as follows:
6293 10 =6 10 3 + 2 10 2 + 9 10 1 + 3 10 0
Examples of positional number systems:
· Binary (or base 2) is a positive integer positional (place) number system that allows different numerical values to be represented using two symbols. Most often these are 0 and 1.
· Octal is a positional integer number system based on base 8. It uses the digits 0 to 7 to represent numbers. Octal is often used in areas involving digital devices. Previously, it was widely used in programming and computer documentation, but has now been almost completely replaced by hexadecimal.
· The decimal number system is a positional number system based on integer base 10. The most common number system in the world. The most commonly used symbols for writing numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called Arabic numerals.
· Duodecimal (widely used in ancient times, in some particular areas it is still used now) - a positional number system with an integer base 12. The numbers used are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B. Some peoples of Nigeria and Tibet still use the duodecimal number system, but echoes of it can be found in almost every culture. In Russian there is the word "dozen", in English "dozen", in some places the word twelve is used instead of "ten", as a round number, for example, wait 12 minutes.
· Hexadecimal (most common in programming, as well as in fonts) is a positional number system based on integer base 16. Typically, decimal digits from 0 to 9 are used as hexadecimal digits, and Latin letters from A to F are used to represent numbers from 10 to 15. Widely used in low-level programming and in computer documentation in general, since in modern computers the minimum unit of memory is an 8-bit byte, the values of which are conveniently written in two hexadecimal digits.
· Hexadecimal (measurement of angles and, in particular, longitude and latitude) is a positional number system based on the integer base 60. Used in ancient times in the Middle East. The consequences of this number system are the division of angular and arc degrees (as well as hours) into 60 minutes and minutes into 60 seconds.
The greatest interest when working on a computer is the number systems with bases 2, 8 and 16. These number systems are usually enough for the full-fledged work of both a person and a computer, but sometimes, due to various circumstances, you still have to turn to other number systems, for example to ternary, septal or base 32 number systems.
To operate with numbers written in such non-traditional systems, you need to keep in mind that they are fundamentally no different from the usual decimal system. Addition, subtraction, and multiplication in them are carried out according to the same scheme.
Other number systems are not used mainly because Everyday life people are accustomed to using the decimal number system, and no other is required. In computers, the binary number system is used, since it is quite simple to operate with numbers written in binary form.
The hexadecimal system is often used in computer science, since writing numbers in it is much shorter than writing numbers in the binary system. The question may arise: why not use a number system, for example base 50, to write very large numbers? Such a number system requires 10 ordinary digits plus 40 signs, which would correspond to the numbers from 10 to 49, and it is unlikely that anyone would like to work with these forty characters. Therefore in real life Number systems based on bases greater than 16 are practically not used.
Introduction to Fractals
Logarithmic function in problems
Example43. Solve the system of equations Solution Let's transform the second equation into the system, applying the definition of the logarithm and taking into account that the expression under the logarithm sign must be strictly positive: Answer: . Example 44...
Positional games
Positional games
Designing mathematics lessons on the topic "Numbering" using modern teaching tools
The positional number system first appeared in ancient Babylon. In India, the system works in the form of positional decimal numbering using zero; the Arab nation borrowed this number system from the Indians, and from them, in turn...
A number system is a way of recording (representing) numbers. The various number systems that existed before and that are currently in use are divided into two groups: · positional, · non-positional...
Notation. Recording actions on numbers
The various number systems that existed in the past and that are used today can be divided into non-positional and positional. The signs used to write numbers are called digits...
Notation. Recording actions on numbers
The binary number system was invented by mathematicians and philosophers even before the advent of computers (XVII - XIX centuries). Some ideas behind binary system, were essentially known in ancient China...
Notation. Recording actions on numbers
The most common number systems are binary, hexadecimal and decimal and octal...
1.1 The history of the emergence of various number systems Primitive man almost did not have to count. “One”, “two” and “many” - these are all his numbers. But we - modern people - have to deal with numbers literally at every step...
Number systems and basics of binary encodings
In the most ancient numbering, only the sign “|” was used. for one, and each natural number was written by repeating the unit symbol as many times as there are units in that number...
Number systems and basics of binary encodings
In addition to the decimal number system, positional number systems with any other natural base are possible. In different historical periods, many peoples widely used different number systems...
Number systems and basics of binary encodings
1.5.1 Addition and subtraction In the system with the base i, the numbers 0, 1, 2, ..., c - 1 are used to denote zero and the first c-1 of natural numbers. To perform the operation of addition and subtraction, a table is compiled for adding single-digit numbers.. .
Number systems and basics of binary encodings
The decimal system so familiar to us turned out to be inconvenient for computers. If in mechanical computing devices that use the decimal system, it is enough to simply use a multi-state element (a wheel with nine teeth) ...
Fractals - a new branch of mathematics
The concept of L-systems, closely related to self-similar fractals, appeared only in 1968 thanks to Aristrid Lindenmayer. Initially, L-systems were introduced in the study of formal languages...
While studying encodings, I realized that I did not understand number systems well enough. Nevertheless, I often used 2-, 8-, 10-, 16-th systems, converted one to another, but everything was done “automatically”. Having read many publications, I was surprised by the lack of a single one written in simple language, articles on such basic material. That is why I decided to write my own, in which I tried to present the basics of number systems in an accessible and orderly manner.
Introduction
Notation is a way of recording (representing) numbers.What does this mean? For example, you see several trees in front of you. Your task is to count them. To do this, you can bend your fingers, make notches on a stone (one tree - one finger/notch), or match 10 trees with an object, for example, a stone, and a single specimen with a stick, and place them on the ground as you count. In the first case, the number is represented as a string of bent fingers or notches, in the second - a composition of stones and sticks, where stones are on the left and sticks on the right
Number systems are divided into positional and non-positional, and positional, in turn, into homogeneous and mixed.
Non-positional- the most ancient, in it each digit of a number has a value that does not depend on its position (digit). That is, if you have 5 lines, then the number is also 5, since each line, regardless of its place in the line, corresponds to only 1 item.
Positional system- the meaning of each digit depends on its position (digit) in the number. For example, the 10th number system that is familiar to us is positional. Let's consider the number 453. The number 4 indicates the number of hundreds and corresponds to the number 400, 5 - the number of tens and is similar to the value 50, and 3 - units and the value 3. As you can see, the larger the digit, the higher the value. The final number can be represented as the sum 400+50+3=453.
Homogeneous system- for all digits (positions) of a number the set of valid characters (digits) is the same. As an example, let's take the previously mentioned 10th system. When writing a number in a homogeneous 10th system, you can use only one digit from 0 to 9 in each digit, thus the number 450 is allowed (1st digit - 0, 2nd - 5, 3rd - 4), but 4F5 is not, because the character F is not included in the set of numbers 0 to 9.
Mixed system- in each digit (position) of a number, the set of valid characters (digits) may differ from the sets of other digits. A striking example is the time measurement system. In the category of seconds and minutes there are 60 different symbols possible (from “00” to “59”), in the category of hours – 24 different symbols (from “00” to “23”), in the category of day – 365, etc.
Non-positional systems
As soon as people learned to count, the need to write down numbers arose. In the beginning, everything was simple - a notch or dash on some surface corresponded to one object, for example, one fruit. This is how the first number system appeared - unit.Unit number system
A number in this number system is a string of dashes (sticks), the number of which is equal to the value of the given number. Thus, a harvest of 100 dates will be equal to a number consisting of 100 dashes.But this system has obvious inconveniences - the larger the number, the more longer line from sticks. In addition, you can easily make a mistake when writing a number by accidentally adding an extra stick or, conversely, not writing it down.
For convenience, people began to group sticks into 3, 5, and 10 pieces. At the same time, each group corresponded to a specific sign or object. Initially, fingers were used for counting, so the first signs appeared for groups of 5 and 10 pieces (units). All this made it possible to create more convenient systems for recording numbers.
Ancient Egyptian decimal system
In Ancient Egypt, special symbols (numbers) were used to represent the numbers 1, 10, 10 2, 10 3, 10 4, 10 5, 10 6, 10 7. Here are some of them:Why is it called decimal? As stated above, people began to group symbols. In Egypt, they chose a grouping of 10, leaving the number “1” unchanged. IN in this case, the number 10 is called the base decimal system, and each symbol is a representation of the number 10 to some degree.
Numbers in the ancient Egyptian number system were written as a combination of these
characters, each of which was repeated no more than nine times. The final value was equal to the sum of the elements of the number. It is worth noting that this method of obtaining a value is characteristic of every non-positional number system. An example would be the number 345:
Babylonian sexagesimal system
Unlike the Egyptian system, the Babylonian system used only 2 symbols: a “straight” wedge to indicate units and a “recumbent” wedge to indicate tens. To determine the value of a number, you need to divide the image of the number into digits from right to left. A new discharge begins with the appearance of a straight wedge after a recumbent one. Let's take the number 32 as an example:
The number 60 and all its powers are also denoted by a straight wedge, like “1”. Therefore, the Babylonian number system was called sexagesimal.
The Babylonians wrote all numbers from 1 to 59 in a decimal non-positional system, and large values in a positional system with a base of 60. Number 92:
The recording of the number was ambiguous, since there was no digit indicating zero. The representation of the number 92 could mean not only 92=60+32, but also, for example, 3632=3600+32. To determine the absolute value of a number, special character to indicate a missing sexagesimal digit, which corresponds to the appearance of the digit 0 in the decimal number notation:
Now the number 3632 should be written as:
The Babylonian sexagesimal system is the first number system based in part on the positional principle. This number system is still used today, for example, when determining time - an hour consists of 60 minutes, and a minute consists of 60 seconds.
Roman system
The Roman system is not very different from the Egyptian one. It uses capital Latin letters I, V, X, L, C, D and M to represent the numbers 1, 5, 10, 50, 100, 500 and 1000, respectively. A number in the Roman numeral system is a set of consecutive digits.Methods for determining the value of a number:
- The value of a number is equal to the sum of the values of its digits. For example, the number 32 in the Roman numeral system is XXXII=(X+X+X)+(I+I)=30+2=32
- If there is a smaller one to the left of the larger digit, then the value is equal to the difference between the larger and smaller digits. At the same time, the left digit can be less than the right one by a maximum of one order of magnitude: for example, only X(10) can appear before L(50) and C(100) among the “lowest” ones, and only before D(500) and M(1000) C(100), before V(5) - only I(1); the number 444 in the number system under consideration will be written as CDXLIV = (D-C)+(L-X)+(V-I) = 400+40+4=444.
- The value is equal to the sum of the values of groups and numbers that do not fit into points 1 and 2.
1) Slavic
2) Greek (Ionian)
Positional number systems
As mentioned above, the first prerequisites for the emergence of a positional system arose in ancient Babylon. In India, the system took the form of positional decimal numbering using zero, and from the Indians this number system was borrowed by the Arabs, from whom the Europeans adopted it. For some reason, in Europe the name “Arab” was assigned to this system.Decimal number system
This is one of the most common number systems. This is what we use when we name the price of a product and say the bus number. Each digit (position) can only use one digit from the range from 0 to 9. The base of the system is the number 10.For example, let’s take the number 503. If this number were written in a non-positional system, then its value would be 5+0+3 = 8. But we have a positional system and that means each digit of the number must be multiplied by the base of the system, in this case the number “ 10”, raised to a power equal to the digit number. It turns out that the value is 5*10 2 + 0*10 1 + 3*10 0 = 500+0+3 = 503. To avoid confusion when working with several number systems simultaneously, the base is indicated as a subscript. Thus, 503 = 503 10.
In addition to the decimal system, the 2-, 8-, and 16th systems deserve special attention.
Binary number system
This system is mainly used in computing. Why didn't they use the usual 10th? The first computer was created by Blaise Pascal, who used a decimal system in it, which turned out to be inconvenient in modern electronic machines, since it required the production of devices capable of operating in 10 states, which increased their price and the final dimensions of the machine. Elements operating in the 2nd system do not have these shortcomings. However, the system in question was created long before the invention of computers and has its “roots” in the Incan civilization, where quipus were used - complex rope weaves and knots.The binary positional number system has a base of 2 and uses 2 symbols (digits) to write numbers: 0 and 1. Only one digit is allowed in each digit - either 0 or 1.
An example is the number 101. It is similar to the number 5 in the decimal number system. In order to convert from 2 to 10, you need to multiply each digit of a binary number by the base “2” raised to a power equal to the place value. Thus, the number 101 2 = 1*2 2 + 0*2 1 + 1*2 0 = 4+0+1 = 5 10.
Well, for machines the 2nd number system is more convenient, but we often see and use numbers in the 10th system on the computer. How then does the machine determine what number the user is entering? How does it translate a number from one system to another, since it only has 2 symbols - 0 and 1?
In order for a computer to work with binary numbers (codes), they must be stored somewhere. To store each individual digit, a trigger is used, which is electronic circuit. It can be in 2 states, one of which corresponds to zero, the other to one. To remember a single number, a register is used - a group of triggers, the number of which corresponds to the number of digits in a binary number. And the set of registers is RAM. The number contained in the register is a machine word. Arithmetic and logical operations with words are performed by an arithmetic logic unit (ALU). To simplify access to registers, they are numbered. The number is called the register address. For example, if you need to add 2 numbers, it is enough to indicate the numbers of the cells (registers) in which they are located, and not the numbers themselves. Addresses are written in octal and hexadecimal systems (they will be discussed below), since the transition from them to the binary system and back is quite simple. To transfer from the 2nd to the 8th, the number must be divided into groups of 3 digits from right to left, and to move to the 16th - 4. If there are not enough digits in the leftmost group of digits, then they are filled from the left with zeros, which are called leading. Let's take the number 101100 2 as an example. In octal it is 101 100 = 54 8, and in hexadecimal it is 0010 1100 = 2C 16. Great, but why do we see decimal numbers and letters on the screen? When you press a key, a certain sequence of electrical impulses is transmitted to the computer, and each symbol has its own sequence of electrical impulses (zeros and ones). The keyboard and screen driver program accesses code table characters (for example, Unicode, which allows you to encode 65536 characters), determines which character the received code corresponds to and displays it on the screen. Thus, texts and numbers are stored in computer memory in binary code, and programmatically are converted into images on the screen.
Octal number system
The 8th number system, like the binary one, is often used in digital technology. It has a base of 8 and uses the digits 0 to 7 to write numbers.An example of an octal number: 254. To convert to the 10th system, each digit of the original number must be multiplied by 8 n, where n is the digit number. It turns out that 254 8 = 2*8 2 + 5*8 1 + 4*8 0 = 128+40+4 = 172 10.
Hexadecimal number system
The hexadecimal system is widely used in modern computers, for example, it is used to indicate color: #FFFFFF - white. The system in question has a base of 16 and uses the following numbers to write: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B. C, D, E, F, where the letters are 10, 11, 12, 13, 14, 15 respectively.Let's take the number 4F5 16 as an example. To convert to the octal system, we first convert the hexadecimal number into binary, and then, dividing it into groups of 3 digits, into octal. To convert a number to 2, you need to represent each digit as a 4-bit binary number. 4F5 16 = (100 1111 101) 2 . But in groups 1 and 3 there is not enough digit, so let’s fill each with leading zeros: 0100 1111 0101. Now you need to divide the resulting number into groups of 3 digits from right to left: 0100 1111 0101 = 010 011 110 101. Let’s convert each binary group to the octal system, multiplying each digit by 2 n, where n is the digit number: (0*2 2 +1*2 1 +0*2 0) (0*2 2 +1*2 1 +1*2 0) (1*2 2 +1*2 1 +0*2 0) (1*2 2 +0*2 1 +1*2 0) = 2365 8 .
In addition to the considered positional number systems, there are others, for example:
1) Trinity
2) Quaternary
3) Duodecimal
Positional systems are divided into homogeneous and mixed.
Homogeneous positional number systems
The definition given at the beginning of the article describes homogeneous systems quite fully, so clarification is unnecessary.Mixed number systems
To the already given definition we can add the theorem: “if P=Q n (P,Q,n are positive integers, while P and Q are bases), then the recording of any number in the mixed (P-Q) number system identically coincides with writing the same number in the number system with the base Q.”Based on the theorem, we can formulate rules for transferring from P to Q-th system and vice versa:
- To convert from Q to P, you need a number in Q system, divide into groups of n digits, starting with the right digit, and replace each group with one digit in the P-th system.
- To convert from P-th to Q-th, it is necessary to convert each digit of a number in the P-th system to Q-th and fill the missing digits with leading zeros, with the exception of the left one, so that each number in the system with base Q consists of n digits .
Mixed number systems are also, for example:
1) Factorial
2) Fibonacci
Conversion from one number system to another
Sometimes you need to convert a number from one number system to another, so let's look at ways to convert between different systems.Conversion to decimal number system
There is a number a 1 a 2 a 3 in the number system with base b. To convert to the 10th system, it is necessary to multiply each digit of the number by b n, where n is the number of the digit. Thus, (a 1 a 2 a 3) b = (a 1 *b 2 + a 2 *b 1 + a 3 *b 0) 10.Example: 101 2 = 1*2 2 + 0*2 1 + 1*2 0 = 4+0+1 = 5 10
Conversion from decimal number system to others
Whole part:- We successively divide the integer part of the decimal number by the base of the system into which we are converting until the decimal number equals zero.
- The remainders obtained during division are the digits of the desired number. Number in new system write down starting from the last remainder.
- We multiply the fractional part of the decimal number by the base of the system to which we want to convert. Separate the whole part. We continue to multiply the fractional part by the base of the new system until it equals 0.
- Numbers in the new system are made up of whole parts of multiplication results in the order corresponding to their production.
15\8 = 1, remainder 7
1\8 = 0, remainder 1
Having written all the remainders from bottom to top, we get the final number 17. Therefore, 15 10 = 17 8.
Converting from binary to octal and hexadecimal
To convert to octal, we divide the binary number into groups of 3 digits from right to left, and fill the missing outermost digits with leading zeros. Next, we transform each group by multiplying the digits sequentially by 2n, where n is the number of the digit.Let's take the number 1001 2 as an example: 1001 2 = 001 001 = (0*2 2 + 0*2 1 + 1*2 0) (0*2 2 + 0*2 1 + 1*2 0) = (0+ 0+1) (0+0+1) = 11 8
To convert to hexadecimal, we divide the binary number into groups of 4 digits from right to left, then similar to the conversion from 2nd to 8th.
Convert from octal and hexadecimal to binary
Conversion from octal to binary - we convert each digit of an octal number into a binary 3-digit number by dividing by 2 (for more information about division, see the paragraph “Converting from the decimal number system to others” above), fill the missing outermost digits with leading zeros.For example, consider the number 45 8: 45 = (100) (101) = 100101 2
Translation from the 16th to the 2nd - we convert each digit of a hexadecimal number into a binary 4-digit number by dividing by 2, filling the missing outer digits with leading zeros.
Converting the fractional part of any number system to decimal
The conversion is carried out in the same way as for integer parts, except that the digits of the number are multiplied by the base to the power “-n”, where n starts from 1.
Example: 101,011 2 = (1*2 2 + 0*2 1 + 1*2 0), (0*2 -1 + 1*2 -2 + 1*2 -3) = (5), (0 + 0 .25 + 0.125) = 5.375 10
Converting the fractional part of binary to 8th and 16th
The translation of the fractional part is carried out in the same way as for whole parts of a number, with the only exception that the division into groups of 3 and 4 digits goes to the right of the decimal point, the missing digits are supplemented with zeros to the right.Example: 1001.01 2 = 001 001, 010 = (0*2 2 + 0*2 1 + 1*2 0) (0*2 2 + 0*2 1 + 1*2 0), (0*2 2 + 1*2 1 + 0*2 0) = (0+0+1) (0+0+1), (0+2+0) = 11.2 8
Converting the fractional part of the decimal system to any other
To convert the fractional part of a number to other number systems, you need to turn the whole part into zero and begin multiplying the resulting number by the base of the system to which you want to convert. If, as a result of multiplication, whole parts appear again, they must be turned to zero again, after first remembering (writing down) the value of the resulting whole part. The operation ends when the fractional part is completely zero.For example, let's convert 10.625 10 to binary:
0,625*2 = 1,25
0,250*2 = 0,5
0,5*2 = 1,0
Writing all the remainders from top to bottom, we get 10.625 10 = (1010), (101) = 1010.101 2